Internet Draft Anura P. Jayasumana
Expiration Date: October 2, 2007 Colorado State University
Nischal M. Piratla
Deutsche Telekom Labs
Tarun Banka
Colorado State University
Abhijit A. Bare
Rick Whitner
Jerry McCollom
Agilent Technologies
April 2, 2007
Reorder Density and Reorder Buffer-occupancy Density - Metrics forPacket Reordering Measurementsdraft-jayasumana-reorder-density-07.txt
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Abstract
This document presents two metrics for packet reordering, namely,
Reorder Density (RD) and Reorder Buffer-occupancy Density (RBD). A
threshold is used to clearly define when a packet is considered lost,
to bound computational complexity at O(N), and to keep the memory
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Packet reordering is a phenomena that occurs in Internet Protocol
(IP) networks. Major causes of packet reordering include, but are
not limited to, packet striping at layers 2 and 3 [Ben99,Jai03],
priority scheduling (e.g., Diffserv), and route fluttering
[Pax97,Boh03]. Reordering leads to degradation of the performance of
many applications [Ben99,Bla02,Lao02]. Increased link speeds[Bar04],
increased parallelism within routers and switches, QoS support, and
load balancing among links all point to increased packet reordering
in future networks.
Effective metrics for reordering are required to measure and quantify
reordering. A good metric or a set of metrics capturing the nature of
reordering can be expected to provide insight into the reordering
phenomenon in networks. It may be possible to use such metrics to
predict the effects of reordering on applications that are sensitive
to packet reorder, and perhaps even to compensate for reordering. A
metric for reordered packets, may also help evaluate network
protocols and implementations with respect to their impact on packet
reordering.
The percentage of out-of-order packets is often used as a metric for
characterizing reordering. However, this metric is vague and lacks
in detail. Further, there is no uniform definition for the degree of
reordering of an arrived packet [Ban02,Pi05a]. For example, consider
the two packet sequences (1, 3, 4, 2, 5) and (1, 4, 3, 2, 5). It is
possible to interpret the reordering of packets in these sequences
differently. For example,
(i) Packets 2, 3 and 4 are out-of-order in both cases.
(ii) Only packet 2 is out-of-order in the first sequence, while
packets 2 and 3 are out-of-order in the second.
(iii)Packets 3 and 4 are out-of-order in both the sequences.
(iv) Packets 2, 3 and 4 are out-of-order in the first sequence,
while packets 4 and 2 are out-of-order in the second sequence.
In essence, the percentage of out-of-order packets as a metric of
reordering is subject to interpretation and cannot capture the
reordering unambiguously and hence, accurately.
Other metrics attempt to overcome this ambiguity by defining only the
late packets or only the early packets as being reordered. However,
measuring reordering based only on late or only on early packets is
not always effective. Consider, for example the sequence (1, 20, 2,
3,..,19, 21, 22, ...); the only anomaly is that packet 20 is
delivered immediately after packet 1. A metric based only on
lateness will indicate a high degree of reordering, even though in
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this example it is a single packet arriving ahead of others.
Similarly, a metric based only on earliness does not accurately
capture reordering caused by a late arriving packet. A complete
reorder metric must account for both earliness and lateness, and must
be able to differentiate between the two. The inability to capture
both the earliness and the lateness precludes a metric from being
useful for estimating end-to-end reordering based on reordering in
constituent subnets.
The sensitivity to packet reordering can vary significantly from one
application to the other. Consider again the packet sequence (1, 3,
4, 2, 5). If buffers are available to store packets 3 and 4 while
waiting for packet 2, an application can recover from reordering.
However, with certain real-time applications, the arrival of packet 2
out of order may render it useless. While one can argue that a good
packet reordering measurement scheme should capture
application-specific effects, a counter argument can also be made
that packet reordering should be measured strictly with respect to
the order of delivery independent of the application.
2. Attributes of Packet Reordering Metrics
The first and foremost requirement of a packet reorder metric is its
ability to capture the amount and extent of reordering in a sequence
of packets. The fact that a measure varies with reordering of packets
in a stream does not make it a good metric. In [Ben99], authors have
provided desirable features of a reordering metric. This list encloses
the foremost requirement stated above, simplicity, low sensitivity to
packet loss, ability to combine reorder measures from two networks,
minimal value for in-order data, and independence to data size. These
features are explained below in detail, along with additional desired
features. Note, the ability to combine reorder measures from two
networks is added to broader applicability and data size independence
is discussed under evaluation complexity. However, data size
independence could also refer to the final measure, as in percentage
reordering oR even a normalized representation.
a) Simplicity
An ideal metric is one that is simple to understand and evaluate,
and yet informative, i.e., able to provide a complete picture of
reordering. Percentage of packets reordered is the simplest
singleton metric; But the ambiguity in its definition as discussed
earlier, and its failure to carry the extent of reordering make it
less informative. On the other hand, keeping track of the
displacements of each and every packet without compressing the
data will contain all the information about reordering, but it is
not simple to evaluate or use.
A simpler metric may be preferred in some cases even though it
does not capture reordering completely, while other cases may
demand more complex, yet complete metric.
In striving to strike a balance, the lateness based metrics
consider only the late packets as reordered, and earliness based
metrics only the early packets as reordered. A metric based only
on earliness or only on lateness however captures only a part of
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information associated with reordering. A metric capturing both
early and late arrivals in contrast provides a complete picture of
reordering in a sequence.
b) Low Sensitivity to Packet Loss and Duplication
A reorder metric should treat only an out-of-order packet as
reordered, i.e., if a packet is lost during transit then this
should not result in its following packets, which arrive in order,
to be classified as out of order. Consider the sequence (1, 3, 4,
5, 6). If packet 2 has been lost, the sequence should not be
considered to contain any out-of-order packets. Similarly, if
multiple copies of a packet (duplicates) are delivered, this must
not result in a packet being classified as out of order, as long
as one copy arrives in the proper position. For example, sequence
(1, 2, 3, 2, 4, 5) has no reordering. The lost and duplicate
packet counts may be tracked using metrics specifically to measure
those, e.g., percentage of lost packets, and percentage of
duplicate packets.
c) Low evaluation complexity
Memory and time complexities associated with evaluating a metric
play a vital role in implementation and real-time measurements.
Spatial/memory complexity corresponds to the amount of buffers
required for the overall measurement process, whereas
time/computation complexity refers to the number of computation
steps involved in computing the amount of reordering in
a sequence. On-the-fly evaluation of the metric for large streams
of packets requires the computational complexity to be O(N), where
N denotes the number of received packets, used for reordering
measure. This allows the metric to be updated in constant-time as
each packet arrives. In the absence of a threshold defining losses
or the number of sequence numbers to buffer for detection of
duplicates, the worst-case complexity of loss and duplication
detection will increase with N. The rate of increase will depend
among other things on the value of N and the implementation of
duplicate detection scheme.
d) Robustness
Reorder measurement should be robust against different network
phenomena and peculiarities in measurement or sequences such as a
very late arrival of a duplicate packet, or even a rogue packet
due to an error or sequence number wrap-around. The impact due to
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an event associated with a single or a small number of packets
should have a sense of proportionality on the reorder measure.
Consider for example, the arrival sequence: (1, 5430, 2, 3, 4,
5,..) where packet 5430 appears to be very early; it may be either
due to sequence rollover in test streams or some unknown reason.
e) Broad applicability
A framework for IP performance metrics [RFC2330] states: "The
metrics must aid users and providers in understanding the
performance they experience or provide."
Rather than being a mere value or a set of values that changes
with the reordering of packets in a stream, a reorder metric
should be useful for a variety of purposes. An application or a
transport protocol implementation, for example, may be able to use
the reordering information to allocate resources to recover from
reordering. A metric may be useful for TCP flow control, buffer
resource allocation for recovery from reordering and /or network
diagnosis.
The ability to combine the reorder metrics of constituent subnets
to provide the end to end reordering would be an extremely useful
property. In the absence of this property, no amount of individual
network measurements short of measuring the reordering for the
pair of endpoints of interest would be useful in predicting the
end-to-end reordering.
The ability to provide different types of information based on
monitoring or diagnostic needs also broadens the applicability of
a metric. Examples of applicable information for reordering may
include parameters such as the percentage of reordered packets that
resulted in fast retransmissions in TCP, or the percentage of
utilization of reorder recovery buffer.
3. Reorder Density and Reorder Buffer-occupancy Density
In this memo, we define two discrete density functions, Reorder
Density (RD) and Reorder Buffer-occupancy Density (RBD), that capture
the nature of reordering in a packet stream. These two metrics can
be used individually or collectively to characterize the reordering
in a packet stream. Also presented are algorithms for real-time
evaluation of these metrics for an incoming packet stream.
RD is defined as the distribution of displacements of packets from
their original positions, normalized with respect to the number of
packets. An early packet corresponds to a negative displacement and
a late packet to a positive displacement. A threshold on displacement
is used to keep the computation within bounds. Choice of such a
threshold is important to the needs of measurement and is further
discussed in Section 5. In other terms, if user Duplicate packets
are accounted for when evaluating these displacements.
The ability of RD to capture the nature and properties of reordering
in a comprehensive manner has been demonstrated in [Pi05a, Pi05b,
Pi05c,Pi07]. The RD observed at the output port of a subnet when the
input is an in-order packet stream, can be viewed as a "reorder
response" of a network, a concept somewhat similar to the "system
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response" or "impulse response" used in traditional system theory.
For a subnet under stationary conditions, RD is the probability
density of the packet displacement. RD measured on individual
subnets can be combined, using the convolution operation, to
predict the end-to-end reorder characteristics of the network formed
by the cascade of subnets under a fairly broad set of conditions
[Pi05b]. RD also shows significant promise as a tool for analytical
modeling of reordering, as demonstrated with a load-balancing
scenario in [Pi06]. Use of a threshold to define the condition under
which a packet is considered lost, makes the metric robust, efficient
and adaptable for different network and stream characteristics.
RBD is the normalized histogram of the occupancy of a hypothetical
buffer that would allow the recovery from out-of-order delivery of
packets. If an arriving packet is early, it is added to a
hypothetical buffer until it can be released in order [Ban02]. The
occupancy of this buffer after each arrival is used as the measure of
reordering. A threshold, used to declare a packet as lost, keeps the
complexity of computation within bounds. The threshold may be
selected based on application requirements in situations where the
late arrival of a packet makes it useless, e.g., a real-time
applications. In [Ban02], this metric was called RD and buffer
occupancy was known as displacement.
RD and RBD are simple, yet useful, metrics that for measurement and
evaluation of reordering. These metrics are robust against many
peculiarities, such as those discussed previously, and have a
computational complexity of O(N), where N is the received sequence
size. RD is orthogonal to loss and duplication, whereas RBD is
orthogonal to duplication.
A detailed comparison of these and other proposed metrics for
reordering is presented in [Pi07].
The following terms are used to formally define RD, RBD, and the
measurement algorithms. Wraparound of sequence numbers is not
explicitly addressed in this document, but with the use of modulo-N
arithmetic, all claims made here remain valid in the presence of
wraparound.
3.1 Receive_index (RI)
Consider a sequence of packets (1, 2, ..., N) transmitted over a
network. A receive_index RI (1, 2, ...), is a value assigned
to a packet as it arrives at its destination according to the order
of arrival. A receive_index is not assigned to duplicate packets,
and the receive_index value skips the value corresponding to a lost
packet. (The detection of loss and duplication for this purpose is
described in section 6.) In the absence of reordering, the sequence
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number of the packet and the receive_index are the same for each
packet.
RI is used to compute earliness and lateness of an arriving packet.
Below are two examples of received sequences with receive_index
values for a sequence of 5 packets (1, 2, 3, 4, 5) arriving out of
order:
Example 1:
Arrived sequence: 2 1 4 5 3
Receive_index: 1 2 3 4 5
Example 2:
Arrived sequence: 1 4 3 5 3
Receive_index: 1 3 4 5 -
In Example 1, there is no loss or duplication. In Example 2, the
packet with sequence number 2 is lost, thus 2 is not assigned as an
RI; packet 3 is duplicated, thus the second copy is not assigned an
RI.
3.2 Out-of-Order Packet
When the sequence number of a packet is not equal to the RI assigned
to it, it is considered an out-of-order packet. Duplicates for which
an RI is not defined are ignored.
3.3 Displacement (D)
Displacement (D) of a packet is defined as the difference between RI
and the sequence number of the packet, i.e., the displacement of
packet i is RI[i] - i. Thus, a negative displacement indicates the
earliness of a packet and a positive displacement to the lateness.
In example 3 below, an arrived sequence with displacements of each
packet is illustrated.
Example 3:
Arrived sequence: 1 4 3 5 3 8 7 6
Receive_index: 1 3 4 5 - 6 7 8
Displacement: 0 -1 1 0 - -2 0 2
3.4 Displacement Threshold (DT)
The displacement threshold is a threshold on the displacement of
packets that allows the metric to classify a packet as lost or
duplicate. Determining when to classify a packet as lost is
difficult because there is no point in time at which a packet can
definitely be classified as lost; the packet may still arrive after
some arbitrarily long delay. However, from a practical point of
view, a packet may be classified as lost if it has not arrived within
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a certain administratively defined displacement threshold, DT.
Similarly, to identify a duplicate packet, it is theoretically
necessary to keep track of all the arrived (or missing) packets.
Again, however, from a practical point of view, missing packets
within a certain window of sequence numbers suffice. Thus, DT is
used as a practical means for declaring a packet as lost or
duplicated. DT makes the metric more robust, keeps the
computational complexity for long sequences within O(N), and keeps
storage requirements independent of N.
If DT is selected too small, reordered packets might be classified as
lost. A large DT will increase both the size of memory required to
keep track of sequence numbers and the length of computation time
required to evaluate the metric. Indeed, it is possible to use two
different thresholds for the two cases. The selection of DT is
further discussed in section 5.
3.5 Displacement Frequency (FD)
Displacement Frequency FD[k] is the number of arrived packets having
a displacement of k, where k takes values from -DT to DT.
3.6 Reorder Density (RD)
RD is defined as the distribution of the Displacement Frequencies
FD[k], normalized with respect to N', where N'is the length of the
received sequence, ignoring lost and duplicate packets. N' is equal
to the sum(FD[k]) for k in [-DT, DT].
3.7 Expected Packet (E)
A packet with sequence number E is expected if E is the largest
number such that all the packets with sequence numbers less than E
have already arrived or have been determined to be lost.
3.8 Buffer Occupancy (B)
An arrived packet with a sequence number greater than that of an
expected packet is considered to be stored in a hypothetical buffer
sufficiently long to permit recovery from reordering. At any packet
arrival instant, the buffer occupancy is equal to the number of
out-of-order packets in the buffer, including the newly arrived
packet. One buffer location is assumed for each packet, although it
is possible to extend the concept to the case where the number of
bytes is used for buffer occupancy. For example, consider the
sequence of packets (1, 2, 4, 5, 3) with expected order (1, 2, 3, 4,
5). When packet 4 arrives the buffer occupancy is 1 because packet 4
arrived early. Similarly, the buffer occupancy becomes 2 when packet
5 arrives. When packet 3 arrives, recovery from reordering occurs
and the buffer occupancy reduces to zero.
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Buffer occupancy threshold is a threshold on the maximum size of the
hypothetical buffer that is used for recovery from reordering. As
with the case of DT for RD, BT is used for loss and duplication
classification for Reorder Buffer-occupancy Density (RBD) computation
(see section 3.11). BT provides robustness, and limits the
computation complexity of RBD.
3.10 Buffer Occupancy Frequency (FB)
At the arrival of each packet the buffer occupancy may take any value
k ranging from 0 to BT. The buffer occupancy frequency FB[k] is the
number of arrival instances after which the occupancy takes the value
of k.
3.11 Reorder Buffer-Occupancy Density (RBD)
Reorder buffer-occupancy density is the buffer occupancy frequencies
normalized by the total number of non-duplicate packets, i.e.,
RBD[k] = FB[k]/N' where N' is the length of the received sequence,
ignoring excessively delayed (deemed lost) and duplicate packets. N'
is also the sum(FB[k]) for all k such that k belongs to [0, BT].
4. Representation of Packet Reordering and Reorder Density
Consider a sequence of packets (1, 2, ..., N). Let the RI assigned
to packet m be "the sequence number m plus an offset dm," i.e.,
RI = m + dm
D = dm
A reorder event of packet m is represented by r(m, dm).
When dm is not equal to zero, a reorder event is said to have
occurred. A packet is late if dm > 0 and early if dm < 0.
Thus, packet reordering of a sequence of packets is completely
represented by the union of reorder events, R, referred to as the
reorder set:
R = {r(m,dm)| dm not equal to 0 for all m}
If there is no reordering in a packet sequence then R is the null
set.
Examples 4 and 5 illustrate the reorder set:
Example 4. No losses or duplicates
Arrived Sequence 1 2 3 5 4 6
Receive_index (RI) 1 2 3 4 5 6
Displacement (D) 0 0 0 -1 1 0
R = {(4,1), (5,-1)}
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Example 5. Packet 4 is lost and 2 is duplicated
Arrived Sequence 1 2 5 3 6 2
Receive_index (RI) 1 2 3 5 6 -
Displacement (D) 0 0 -2 2 0 -
R = {(3, 2), (5, -2)}
RD is defined as the discrete density of the frequency of packets
with respect to their displacements, i.e., the lateness and earliness
from the original position. Let S[k] denote the set of reorder
events in R with displacement equal to k, i.e.,
S[k]= {r(m, dm)| dm = k}
Let |S[k]| be the cardinality of set S[k]. Thus, RD[k] is defined as
|S[k]| normalized with respect to the total number of received
packets (N'). Note that N' does not include duplicates or lost
packets.
RD[k] = |S[k]| / N' for k not equal to zero.
RD[0] corresponds to the packets for which RI is the same as the
sequence number:
RD[0] = 1 - sum(|S[k]| / N')
As defined previously, FD[k] is the measure that keeps track of
|S[k]|.
5. Selection of DT
Although assigning a threshold for determining lost and duplicate
packets might appear to introduce error into the reorder metrics, in
practice this need not be the case. Applications, protocols, and the
network itself operate within finite resource constraints which
introduce practical limits beyond which the choice of certain values
become irrelevant. If the operational nature of an application
is such that a DT can be defined, then using DT in the computation of
reorder metrics will not invalidate nor limit the effectiveness of the
metrics, i.e., increasing DT does not provide any benefit. In the
case of TCP, the transmit and receive window sizes impose a natural
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limit on the useful value of DT. Sequence number wraparound may
provide a useful upper bound for DT in some instances.
If there are no operational constraints imposed by factors as
described above, or if one is purely interested in a more complete
picture of reordering, then DT can be made as large as required. If
DT is equal to the length of the packet sequence (worst case
scenario), a complete picture of reordering is seen. Any metric that
does not rely on a threshold to declare a packet as lost, implicitly
makes one of two assumptions: a) A missing packet is not considered
lost until the end of the sequence, or b) the packet is considered
lost until it arrives. Former corresponds to the case where DT is
set to the length of the sequence. Latter leads to many problems
related to complexity and robustness.
6. Detection of Lost and Duplicate Packets
In RD, a packet is considered lost if it is late beyond DT.
Non-duplicate arriving packets do not have a copy in the buffer and
do not have a sequence number less (earlier) than E. In RBD, a
packet is considered lost if the buffer is filled to its threshold
BT. A packet is considered a duplicate when the sequence number is
less than the expected packet, or if the sequence number is already
in the buffer.
Since RI skips the sequence number of a lost packet, the question
arises as to how to assign an RI to subsequent packets that arrive
before it is known that the packet is lost. This problem arises only
when reorder metrics are calculated in real-time for an incoming
sequence, and not with offline computations. This concern can be
handled in one of two ways:
a) Go-back Method: RD is computed as packets arrive. When a packet
is deemed lost, RI values are corrected and displacements recomputed.
The Go-back Method is only invoked when a packet is lost, and
re-computing involves at most DT packets.
b) Stay-back Method: RD evaluation lags the arriving packets so that
the correct RI and E values can be assigned to each packet as it
arrives. Here, RI is assigned to a packet only once, and the value
assigned is guaranteed to be correct. In the worst case, the
computation lags arriving packet by DT. The lag associated with the
Stay-back Method is incurred only when a packet is missing.
Another issue related to a metric and its implementation is the
robustness against peculiarities that may occur in a sequence as
discussed in Section 2. Consider for example, the arrival sequence:
(1, 5430, 2, 3, 4, 5,...). With RD, a sense of proportionality is
maintained easily using the concept of threshold (DT) and to limit
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the effect of a rogue packet to this threshold. With RD, for
example, as its displacement is greater than threshold, it is
discarded. This way the impact due to the rogue packet, 5430, is
limited at most to DT packets, thus imposing a limit on the amount of
error it can cause in results. Note also that a threshold different
from DT can be used for this purpose. By limiting the time a packet
to remain in the buffer according to a prespecified threshold, RBD
can be made robust against rogue packets as well.
7. Algorithms to evaluate RD and RBD
The algorithms to compute RD and RBD are given below. These
algorithms are applicable for on-line computation of an incoming
packet stream, and provide an up-to-date metric for the packet stream
so far. For simplicity, the sequence numbers are considered to start
from 1 and continue in increments of 1. Only the Stay-back Method of
loss detection is presented here, hence the RD values lag by a
maximum of DT. Algorithm for go-back method is given in [Bar04]. Perl
scripts for these algorithms are posted in [Per04].
7.1 Algorithm for RD
Variables used:
-------------------------------------------------------------------
RI: receive_index.
S: Arrival under consideration for lateness/earliness computation.
D: Lateness or earliness of the packet being processed: dm for m.
FD[ -DT..DT]: Frequency of lateness and earliness.
window[1..DT+1]: List of incoming sequence numbers.
buffer[1..DT]: Array to hold sequence numbers of early arrivals.
window[] and buffer[] are empty at the beginning.
===================================================================
Step 1. Initialize:
Store first unique DT+1 sequence numbers in arriving order into
window;
RI = 1;
Step 2. Repeat (until window is empty):
If (window or buffer contains sequence number RI)
{
Copy first sequence number in window to S;
Delete first sequence number from window;
D = RI - S; # compute displacement
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Tarun Banka <Tarun.Banka@colostate.edu>
Computer Networking Research Laboratory,
Department of Electrical and Computer Engineering,
1373 Colorado State University,
Fort Collins, CO 80523, USA
Abhijit A. Bare <abhijit_bare@agilent.com>
Rick Whitner <rick_whitner@agilent.com>
Jerry McCollom <jerry_mccollom@agilent.com>
Agilent Technologies, 4380 Ziegler Rd.,
Fort Collins, CO 80525, USA
Expiration Date: April 2, 2007
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Jayasumana, et al. Expires October 2, 2007 [Page 23]